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Source: RUDN MATH OLYMP 2024 problem 4

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#1 h Apr 17, 2025, 11:31 AM • 1 Y

Y by FFA21

Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.

This post has been edited 1 time. Last edited by c00lb0y, Apr 18, 2025, 7:13 AM
Reason: unsolved yet!

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#2 h Apr 19, 2025, 10:19 AM • 1 Y

Y by FFA21

Your post is badly written.
"The condition" is -I think so- : for every $t$ , $\det(X(t))=0$ .
"There are no solutions..." ; you mean : "there are no maximal solutions...".
The considered property is false for $n=1$ and true for $n\geq 2$ .
For $n\geq 2$ , choose $X(t)=diag(\dfrac{-1}{t+C},0_{n-1})$ .
EDIT.
For $n=2$ , there are non-diagonal solutions with $2$ constants: $X(t)=\begin{pmatrix}0&\dfrac{\lambda}{t+\mu}\\0&\dfrac{-1}{t+\mu}\end{pmatrix}$ .

This post has been edited 2 times. Last edited by loup blanc, Apr 20, 2025, 2:48 PM

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#3 h May 20, 2025, 9:17 AM

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@loup blanc I don't understand the problem, at the olympiad I gave similar answer as yours, but unfortunately. And I just copied the problem right from olympiad, i don't understand about differentiability and having a solution in some interval

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#4 h May 20, 2025, 12:26 PM

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Here we are only interested in maximal solutions. The maximal solutions given above are defined on $]-\infty,-\mu[$ or $]-\mu,\infty[$ .
In fact, I don't understand the sentence "It is known that there are no solutions of this equation defined on a bounded interval".
For $n=3$ , let $X(t)=diag(\dfrac{-1}{t-2},\dfrac{-1}{t-3},0)$ .
Then $X(t)$ is a maximal solution defined on the bounded interval $]2,3[$ .

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